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Localisation and Transport in One-Dimensional Disordered Systems

  • B. Kramer
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 83)

Abstract

The theory of localisation and transport in one-dimensional disordered systems is reviewed. The localisation length is calculated. It is shown that the dc-conductivity vanishes at the absolute zero of the temperature, if the Fermi energy lies in the region of localised states. The Landauer formula for the conductance of a system of finite length is derived. It is shown that the averages of the resistance and the conductance increase, and decrease exponentially with the length of the system, respectively. The distributions of the localisation length, the resistance, and the conductance are discussed. It is shown that the localisation length is the quantity which obeys the central limit theorem, whereas the width of the distributions of the resistance and the conductance increase with increasing size of the system faster than the averages of these quantities in the thermodynamic limit.

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References

  1. 1.
    K. Ishii: Progr. Theor. Phys. (Suppl.) 53, 77 (1973)CrossRefGoogle Scholar
  2. 2.
    A. A. Abrikosov and I. A. Ryzhkin: Adv. Phys. 27, 147 (1978)CrossRefGoogle Scholar
  3. 3.
    A. A. Gogolin: Phys. Reports 86, 2 (1982)CrossRefGoogle Scholar
  4. 4.
    P. Erdös, R. C. Herndon: Adv. Phys. 31, 65 (1982)CrossRefGoogle Scholar
  5. 5.
    G. Czycholl, B. Kramer: Z. Phys. B 39, 193 (1980)CrossRefGoogle Scholar
  6. 6.
    V. I.Oseledec: Trans. Moscow Math. Soc. 19, 197 (1968)Google Scholar
  7. 7.
    J. L. Pichard, G. Sarma: J. Phys. C: Solid State Phys. 14, L127 (1981)CrossRefGoogle Scholar
  8. 8.
    N. F. Mott, W. D. Twose: Adv. Phys. 10, 107 (1961)CrossRefGoogle Scholar
  9. 9.
    S. A. Molcanov: Math. USSR Izvestija 12, 69 (1978)CrossRefGoogle Scholar
  10. 10.
    H. Kunz, B: Souillard: Commun. Math. Phys. 78, 201 (1980)CrossRefGoogle Scholar
  11. 11.
    F. Delyon, H. Kunz, B. Souillard: J. Phys. A: Math. Gen. 16, 25 (1983)CrossRefGoogle Scholar
  12. 12.
    D. J. Thouless: in Ill-condensed Matter, R. Balian, R. Maynard, G. Toulouse, 1 (North-Holland 1979)Google Scholar
  13. 13.
    M. Kappus, F. Wegner: Z. Phys. B45, 15 (1981)CrossRefGoogle Scholar
  14. 14.
    B. Derrida, E. Gardner: J. Physique 45, 1283 (1984)CrossRefGoogle Scholar
  15. 15.
    C. J. Lambert: J. Phys. C: Solid State Phys. 17, 2401 (1984)CrossRefGoogle Scholar
  16. 16.
    R. Johnston, B. Kramer: Z. Phys. B 63, 273 (1986)CrossRefGoogle Scholar
  17. 17.
    M. Kasner, W. Weller: phys. stat. sol. (b) 134, 731 (1986)CrossRefGoogle Scholar
  18. 18.
    A. MacKinnon’ J Phys. C: Solid State Phys. 13, L1031 (1980)CrossRefGoogle Scholar
  19. 19.
    G. Czycholl, B. Kramer, A. MacKinnon: Z. Phys. B 43, 5 (1981)CrossRefGoogle Scholar
  20. 20.
    J. L. Pichard: J. Phys. C: Solid State Phys. 19, 1519 (1986)CrossRefGoogle Scholar
  21. 21.
    D. C. Herbert, R. Jones: J. Phys. C: Solid State Phys. 4, 1145 (1971)CrossRefGoogle Scholar
  22. 22.
    D. J. Thouless, S. Kirkpatrick: J. Phys. C: Solid State Phys. 14, 235 (1981)CrossRefGoogle Scholar
  23. 23.
    R. Landauer: Phil. Mag. 21, 863 (1970)CrossRefGoogle Scholar
  24. 24.
    P. W. Anderson, D. J. Thouless, E. Abrahams, D. S. Fisher: Phys. Rev. B 22, 3519 (1980)CrossRefGoogle Scholar
  25. 25.
    D. C. Langreth, E. Abrahams: Phys. Rev. B 31, 2978 (1981)CrossRefGoogle Scholar
  26. 26.
    D. S. Fisher, P. A. Lee: Phys. Rev. B 23, 6851 (1981)CrossRefGoogle Scholar
  27. 27.
    E. N. Economou, C. M. Soukoulis: Phys. Rev. Letters, 618 (1981); ibid. 47, 972 (1981)CrossRefGoogle Scholar
  28. 28.
    D. J. Thouless: Phys. Rev. Letters, 47, 92 (1981)CrossRefGoogle Scholar
  29. 29.
    P. W. Anderson: Phys. Rev. 23, 4828 (1981)CrossRefGoogle Scholar
  30. 30.
    R. Landauer: Springer Series in Solid-State Sciences 61 (Proceedings of LITPIM), 38 (Springer-Verlag 1985)Google Scholar
  31. 31.
    M. Büttiker, Y. Imry, R. Landauer, S. Pinhas: Phys. Rev. B 31, 6207 (1985)CrossRefGoogle Scholar
  32. 32.
    E. Abrahams, M. Stephen: J. Phys. C: Solid State Phys. 13, L377 (1980)CrossRefGoogle Scholar
  33. 33.
    B. Andereck, E. Abrahams: J. Phys. C: Solid State Phys. 13, L383 (1980)CrossRefGoogle Scholar
  34. 34.
    A. D. Stone, J. D. Joannopoulos, D. J. Chadi: Phys. Rev B 24, 5583 (1981)CrossRefGoogle Scholar
  35. 35.
    R. Kree, A. Schmid: Z. Phys. B 42, 297 (1981)CrossRefGoogle Scholar
  36. 36.
    J. Sak, B. Kramer: Phys. Rev. B 24, 1761 (1981)CrossRefGoogle Scholar
  37. 37.
    P. D. Kirkman, J. B. Pendry: J. Phys. C: Solid State Phys. 17, 4327 (1984)CrossRefGoogle Scholar
  38. 38.
    V. L Mel’nikov: JETP Letters 32, 225 (1981)Google Scholar
  39. 39.
    P. D. Kirkman, J. B. Pendry: J. Phys. C: Solid State Phys. 17, 5707 (1984)CrossRefGoogle Scholar
  40. 40.
    A. J. O’Connor: Commun. Math. Phys., 45, 63 (1975)CrossRefGoogle Scholar
  41. 41.
    A. A. Abrikosov: Solid State Commun. 37, 997 (1981)CrossRefGoogle Scholar
  42. 42.
    Y. Kantor, A. Kapitulnik: Sol. State Commun. 42, 161 (1982)CrossRefGoogle Scholar
  43. 43.
    K. Tankei, F. Takano: J. Phys. Soc. Japan, 55, 3516 (1986)CrossRefGoogle Scholar
  44. 44.
    A. D. Stone, J. D. Joannopoulos: Phys. Rev. B 25, 2400 (1982)CrossRefGoogle Scholar
  45. 45.
    P. A. Lee, A. D. Stone, H. Fukuyama: Phys. Rev. B. 35, 1039 (1987-II)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • B. Kramer
    • 1
  1. 1.Physikalisch-Technische Bundesanstalt BraunschweigBraunschweigFed.Rep.of Germany

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